2.3 BASIC DIFFERENTIATION FORMULAS. dy dx 12 d dx x 1 12 x 2 12

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1 SECTION 2.3 BASIC DIFFERENTIATION FORMULAS BASIC DIFFERENTIATION FORMULAS EXAMLE A At what points on the hyperbola xy 12 is the tangent line parallel to the line 3x y 0? SOLUTION Since xy 12 can be written as y 12x, we have (_2, _6) FIGURE 1 y (2, 6) xy=12 0 x 3x+y=0 Let the x-coordinate of one of the points in question be a. Then the slope of the tangent line at that point is 12a 2. This tangent line will be parallel to the line 3x y 0, or y 3x, if it has the same slope, that is, 3. Equating slopes, we get 12 a 2 dy dx 12 d dx x 1 12x 2 12 x 2 3 or a 2 4 a 2 Therefore, the required points are 2, 6 and 2, 6. The hyperbola and the tangents are shown in Figure 1. or EXAMLE B If a rod or piece of wire is homogeneous, then its linear density is uniform and is defined as the mass per unit length and measured in kilograms per meter. Suppose, however, that the rod is not homogeneous but that its mass measured from its left end to a point x is m f x, as shown in Figure 2. x ml FIGURE 2 This part of the rod has mass ƒ. x x The mass of the part of the rod that lies between x x 1 and x x 2 is given by m f x 2 f x 1, so the average density of that part of the rod is average density m x f x 2 f x 1 x 2 x 1 If we now let x l 0 (that is, x 2 l x 1 ), we are computing the average density over smaller and smaller intervals. The linear density at x 1 is the limit of these average densities as x l 0; that is, the linear density is the rate of change of mass with respect to length. Symbolically, lim x l 0 m x dm dx Thus, the linear density of the rod is the derivative of mass with respect to length. For instance, if m f x sx, where x is measured in meters and m in kilograms, then the average density of the part of the rod given by 1 x 1.2 is m x f 1.2 f s kgm

2 2 SECTION 2.3 BASIC DIFFERENTIATION FORMULAS while the density right at x 1 is dm dx x1 1 2sx 0.50 kgm x1 FIGURE 3 V lay the Video V EXAMLE C A current exists whenever electric charges move. Figure 3 shows part of a wire and electrons moving through a shaded plane surface. If Q is the net charge that passes through this surface during a time period t, then the average current during this time interval is defined as average current Q t Q 2 Q 1 t 2 t 1 If we take the limit of this average current over smaller and smaller time intervals, we get what is called the current I at a given time : t 1 Q I lim t l 0 t dq Thus, the current is the rate at which charge flows through a surface. It is measured in units of charge per unit time (often coulombs per second, called amperes). EXAMLE D A chemical reaction results in the formation of one or more substances (called products) from one or more starting materials (called reactants). For instance, the equation 2H 2 O 2 l 2H 2 O indicates that two molecules of hydrogen and one molecule of oxygen form two molecules of water. Let s consider the reaction A B l C where A and B are the reactants and C is the product. The concentration of a reactant A is the number of moles ( 1 mole molecules) per liter and is 23 denoted by A. The concentration varies during a reaction, so A, B, and C are all functions of time t. The average rate of reaction of the product C over a time interval t 1 t t 2 is C t Ct 2 Ct 1 t 2 t 1 But chemists are more interested in the instantaneous rate of reaction, which is obtained by taking the limit of the average rate of reaction as the time interval t approaches 0: C rate of reaction lim dc t l 0 t Since the concentration of the product increases as the reaction proceeds, the derivative dc will be positive, and so the rate of reaction of C is positive. The concentrations of the reactants, however, decrease during the reaction, so, to make

3 SECTION 2.3 BASIC DIFFERENTIATION FORMULAS 3 the rates of reaction of A and B positive numbers, we put minus signs in front of the derivatives da and db. Since A and B each decrease at the same rate that C increases, we have rate of reaction dc da db More generally, it turns out that for a reaction of the form we have aa bb l cc dd 1 a da 1 b db 1 c dc 1 d dd The rate of reaction can be determined from data and graphical methods. In some cases we can use the rate of reaction to find explicit formulas for the concentrations as functions of time. EXAMLE E One of the quantities of interest in thermodynamics is compressibility. If a given substance is kept at a constant temperature, then its volume V depends on its pressure. We can consider the rate of change of volume with respect to pressure namely, the derivative. As increases, V decreases, so 0. The compressibility is defined by introducing a minus sign and dividing this derivative by the volume V: Thus, measures how fast, per unit volume, the volume of a substance decreases as the pressure on it increases at constant temperature. For instance, the volume V (in cubic meters) of a sample of air at 25C was found to be related to the pressure (in kilopascals) by the equation The rate of change of V with respect to when 50 ka is The compressibility at that pressure is isothermal compressibility 1 V V m3 ka 1 V m 3 kam

4 4 SECTION 2.3 BASIC DIFFERENTIATION FORMULAS EXAMLE F When we consider the flow of blood through a blood vessel, such as a vein or artery, we can model the shape of the blood vessel by a cylindrical tube with radius R and length l as illustrated in Figure 4. R r FIGURE 4 Blood flow in an artery l Because of friction at the walls of the tube, the velocity v of the blood is greatest along the central axis of the tube and decreases as the distance r from the axis increases until v becomes 0 at the wall. The relationship between v and r is given by the law of laminar flow discovered by the French physician Jean-Louis-Marie oiseuille in This law states that 1 v 4l R2 r 2 where is the viscosity of the blood and is the pressure difference between the ends of the tube. If and l are constant, then v is a function of r with domain 0, R. [For more detailed information, see W. Nichols and M. O Rourke (eds.), McDonald s Blood Flow in Arteries: Theoretic, Experimental, and Clinical rinciples, 4th ed. (New York: Oxford University ress, 1998).] The average rate of change of the velocity as we move from r r 1 outward to r r 2 is given by and if we let r l 0, we obtain the velocity gradient, that is, the instantaneous rate of change of velocity with respect to r: Using Equation 1, we obtain v dv velocity gradient lim r l 0 r dr dv dr v r vr 2 vr 1 r 2 r 1 For one of the smaller human arteries we can take, R cm, l 2 cm, and 4000 dynescm 2, which gives v r r 2 At r cm the blood is flowing at a speed of v cms r 0 2r 4l 2l 0.027

5 SECTION 2.3 BASIC DIFFERENTIATION FORMULAS 5 and the velocity gradient at that point is dv dr cmscm r To get a feeling for what this statement means, let s change our units from centimeters to micrometers ( 1 cm 10,000 m). Then the radius of the artery is 80 m. The velocity at the central axis is 11,850 ms, which decreases to 11,110 ms at a distance of r 20 m. The fact that dvdr 74 (ms)m means that, when r 20 m, the velocity is decreasing at a rate of about 74 ms for each micrometer that we proceed away from the center.